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1/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

Bijective proof and generalization of Siladi´ c’s partition theorem

Isaac KONAN

IRIF, Paris Diderot

Alea Days 2019, March 21st

Isaac KONAN Siladi` c’s partition theorem

2/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

Overview

1 A Rogers-Ramanujan type identy : Siladi´

c’s partition theorem Partitions and Rogers-Ramanujan type Identities Siladi´ c’s partition theorem Dousse’s refinement

2 Generalization of Siladi´

c’s theorem Infinite set of primary colors Generalized theorem

3 Bijective map 4 Further analysis on higher degrees

Isaac KONAN Siladi` c’s partition theorem

3/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Partitions and Rogers-Ramanujan type Identities Siladi´ c’s partition theorem Dousse’s refinement

Colored integer partitions

Finite “decreasing” sequence of colored positive integers λ = (λ1, . . . , λs). Example : λ = (4, 2, 1, 1)

Isaac KONAN Siladi` c’s partition theorem

3/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Partitions and Rogers-Ramanujan type Identities Siladi´ c’s partition theorem Dousse’s refinement

Colored integer partitions

Finite “decreasing” sequence of colored positive integers λ = (λ1, . . . , λs). Example : λ = (4, 2, 1, 1)

Isaac KONAN Siladi` c’s partition theorem

3/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Partitions and Rogers-Ramanujan type Identities Siladi´ c’s partition theorem Dousse’s refinement

Colored integer partitions

Finite “decreasing” sequence of colored positive integers λ = (λ1, . . . , λs). Example : λ = (4, 2, 1, 1)

• Parts’length : 4, 2, 1, 1
• Color sequence :

c(λ) =

i ci = red · blue · blue · green

Isaac KONAN Siladi` c’s partition theorem

3/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Partitions and Rogers-Ramanujan type Identities Siladi´ c’s partition theorem Dousse’s refinement

Colored integer partitions

Finite “decreasing” sequence of colored positive integers λ = (λ1, . . . , λs). Example : λ = (4, 2, 1, 1)

• Parts’length : 4, 2, 1, 1
• Color sequence :

c(λ) =

i ci = red · blue · blue · green

• Size : |λ| =

i λi = 4 + 2 + 1 + 1 = 8

Isaac KONAN Siladi` c’s partition theorem

3/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Partitions and Rogers-Ramanujan type Identities Siladi´ c’s partition theorem Dousse’s refinement

Colored integer partitions

Finite “decreasing” sequence of colored positive integers λ = (λ1, . . . , λs). Example : λ = (4, 2, 1, 1)

• Parts’length : 4, 2, 1, 1
• Color sequence :

c(λ) =

i ci = red · blue · blue · green

• Size : |λ| =

i λi = 4 + 2 + 1 + 1 = 8

• Consecutive differences (increments)

λi − λi+1 : 2, 1, 0.

Isaac KONAN Siladi` c’s partition theorem

3/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Partitions and Rogers-Ramanujan type Identities Siladi´ c’s partition theorem Dousse’s refinement

Colored integer partitions

Finite “decreasing” sequence of colored positive integers λ = (λ1, . . . , λs). Example : λ = (4, 2, 1, 1)

• Parts’length : 4, 2, 1, 1
• Color sequence :

c(λ) =

i ci = red · blue · blue · green

• Size : |λ| =

i λi = 4 + 2 + 1 + 1 = 8

• Consecutive differences (increments)

λi − λi+1 : 2, 1, 0. For any set of partitions A, enumeration according to partitions’ size and color sequence : GFA(q) =

• λ∈A

c(λ)q|λ|.

Isaac KONAN Siladi` c’s partition theorem

4/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Partitions and Rogers-Ramanujan type Identities Siladi´ c’s partition theorem Dousse’s refinement

Rogers-Ramanujan type identities

Theorem (RR1919)

Same cardinalities for sets of partitions of size n with:

• consecutive differences at least 2,
• parts congruents to ±1 mod 5.

Isaac KONAN Siladi` c’s partition theorem

4/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Partitions and Rogers-Ramanujan type Identities Siladi´ c’s partition theorem Dousse’s refinement

Rogers-Ramanujan type identities

Theorem (RR1919)

Same cardinalities for sets of partitions of size n with:

• consecutive differences at least 2,
• parts congruents to ±1 mod 5.

Rogers-Ramanujan type identity : equality between two sets of partitions with conditions on respectively :

• consecutive differences,
• parts’congruences .

Isaac KONAN Siladi` c’s partition theorem

4/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Partitions and Rogers-Ramanujan type Identities Siladi´ c’s partition theorem Dousse’s refinement

Rogers-Ramanujan type identities

Theorem (RR1919)

Same cardinalities for sets of partitions of size n with:

• consecutive differences at least 2,
• parts congruents to ±1 mod 5.

Rogers-Ramanujan type identity : equality between two sets of partitions with conditions on respectively :

• consecutive differences,
• parts’congruences .

Example of Euler distinct-odd (1748) : distincts parts ≡ odd parts.

Isaac KONAN Siladi` c’s partition theorem

5/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Partitions and Rogers-Ramanujan type Identities Siladi´ c’s partition theorem Dousse’s refinement

Siladi´ c’s partition theorem (2002)

Theorem

The number of partitions (λ1, . . . , λs) of an integer n into parts λi different from 2, such that λi − λi+1 ≥ 5 and with additional conditions for 5 ≤ λi − λi+1 ≤ 8 according to the table below :

λi − λi+1 λi + λi+1 mod 16 λi − λi+1 λi + λi+1 mod 16 5 ±3 6 0, ±4, 8 7 ±1, ±5, ±7 8 0, ±2, ±6, 8

, is equal to the number of partitions of n into distinct odd parts.

Isaac KONAN Siladi` c’s partition theorem

5/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Partitions and Rogers-Ramanujan type Identities Siladi´ c’s partition theorem Dousse’s refinement

Siladi´ c’s partition theorem (2002)

Theorem

The number of partitions (λ1, . . . , λs) of an integer n into parts λi different from 2, such that λi − λi+1 ≥ 5 and with additional conditions for 5 ≤ λi − λi+1 ≤ 8 according to the table below :

λi − λi+1 λi + λi+1 mod 16 λi − λi+1 λi + λi+1 mod 16 5 ±3 6 0, ±4, 8 7 ±1, ±5, ±7 8 0, ±2, ±6, 8

, is equal to the number of partitions of n into distinct odd parts.

Isaac KONAN Siladi` c’s partition theorem

5/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Partitions and Rogers-Ramanujan type Identities Siladi´ c’s partition theorem Dousse’s refinement

Siladi´ c’s partition theorem (2002)

Theorem

The number of partitions (λ1, . . . , λs) of an integer n into parts λi different from 2, such that λi − λi+1 ≥ 5 and with additional conditions for 5 ≤ λi − λi+1 ≤ 8 according to the table below :

λi − λi+1 λi + λi+1 mod 16 λi − λi+1 λi + λi+1 mod 16 5 ±3 6 0, ±4, 8 7 ±1, ±5, ±7 8 0, ±2, ±6, 8

, is equal to the number of partitions of n into distinct odd parts. Example of n = 15

Isaac KONAN Siladi` c’s partition theorem

5/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Partitions and Rogers-Ramanujan type Identities Siladi´ c’s partition theorem Dousse’s refinement

Siladi´ c’s partition theorem (2002)

Theorem

The number of partitions (λ1, . . . , λs) of an integer n into parts λi different from 2, such that λi − λi+1 ≥ 5 and with additional conditions for 5 ≤ λi − λi+1 ≤ 8 according to the table below :

λi − λi+1 λi + λi+1 mod 16 λi − λi+1 λi + λi+1 mod 16 5 ±3 6 0, ±4, 8 7 ±1, ±5, ±7 8 0, ±2, ±6, 8

, is equal to the number of partitions of n into distinct odd parts. Example of n = 15 First kind : (15), (11, 4), (14, 1), (12, 3).

Isaac KONAN Siladi` c’s partition theorem

5/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Partitions and Rogers-Ramanujan type Identities Siladi´ c’s partition theorem Dousse’s refinement

Siladi´ c’s partition theorem (2002)

Theorem

The number of partitions (λ1, . . . , λs) of an integer n into parts λi different from 2, such that λi − λi+1 ≥ 5 and with additional conditions for 5 ≤ λi − λi+1 ≤ 8 according to the table below :

λi − λi+1 λi + λi+1 mod 16 λi − λi+1 λi + λi+1 mod 16 5 ±3 6 0, ±4, 8 7 ±1, ±5, ±7 8 0, ±2, ±6, 8

, is equal to the number of partitions of n into distinct odd parts. Example of n = 15 First kind : (15), (11, 4), (14, 1), (12, 3). Second kind : (15), (11, 3, 1), (9, 5, 1), (7, 5, 3).

Isaac KONAN Siladi` c’s partition theorem

6/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Partitions and Rogers-Ramanujan type Identities Siladi´ c’s partition theorem Dousse’s refinement

Dousse’s weighted words method

Parts occur in two primary colors a, b and three secondary colors a2, b2, ab, and expression of the minimal differences in terms of colored parts.

part in N 8k 8k + 1 8k + 2 8k + 3 colored part (2k + 1)ab (2k + 1)a (2k + 1)b2 (2k + 1)b part in N 8k + 4 8k + 5 8k + 6 8k + 7 colored part (2k + 2)ab (2k + 2)a (2k + 3)a2 (2k + 2)b

Isaac KONAN Siladi` c’s partition theorem

6/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Partitions and Rogers-Ramanujan type Identities Siladi´ c’s partition theorem Dousse’s refinement

Dousse’s weighted words method

Parts occur in two primary colors a, b and three secondary colors a2, b2, ab, and expression of the minimal differences in terms of colored parts.

part in N 8k 8k + 1 8k + 2 8k + 3 colored part (2k + 1)ab (2k + 1)a (2k + 1)b2 (2k + 1)b part in N 8k + 4 8k + 5 8k + 6 8k + 7 colored part (2k + 2)ab (2k + 2)a (2k + 3)a2 (2k + 2)b

In terms of q-series, from colored integers to natural numbers, we do the dilation (q, a, b) − → (q4, q−3, q−1) .

Isaac KONAN Siladi` c’s partition theorem

7/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Partitions and Rogers-Ramanujan type Identities Siladi´ c’s partition theorem Dousse’s refinement

Minimal differences in terms of colored parts

λi\λi+1

a2

• dd

aodd aeven b2

• dd

bodd beven abodd abeven a2

• dd

4 4 3 4 4 3 4 3 aodd 2 2 3 2 2 3 2 1 aeven 3 3 2 3 3 2 3 2 b2

• dd

2 2 3 4 4 3 2 3 bodd 2 2 1 2 2 3 2 1 beven 1 1 2 3 3 2 1 2 abodd 2 2 3 4 4 3 2 3 abeven 3 3 2 3 3 2 3 2 ·

Isaac KONAN Siladi` c’s partition theorem

7/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Partitions and Rogers-Ramanujan type Identities Siladi´ c’s partition theorem Dousse’s refinement

Minimal differences in terms of colored parts

λi\λi+1

a2

• dd

aodd aeven b2

• dd

bodd beven abodd abeven a2

• dd

4 4 3 4 4 3 4 3 aodd 2 2 3 2 2 3 2 1 aeven 3 3 2 3 3 2 3 2 b2

• dd

2 2 3 4 4 3 2 3 bodd 2 2 1 2 2 3 2 1 beven 1 1 2 3 3 2 1 2 abodd 2 2 3 4 4 3 2 3 abeven 3 3 2 3 3 2 3 2 ·

Isaac KONAN Siladi` c’s partition theorem

7/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Partitions and Rogers-Ramanujan type Identities Siladi´ c’s partition theorem Dousse’s refinement

Minimal differences in terms of colored parts

λi\λi+1

a2

• dd

aodd aeven b2

• dd

bodd beven abodd abeven a2

• dd

4 4 3 4 4 3 4 3 aodd 2 2 3 2 2 3 2 1 aeven 3 3 2 3 3 2 3 2 b2

• dd

2 2 3 4 4 3 2 3 bodd 2 2 1 2 2 3 2 1 beven 1 1 2 3 3 2 1 2 abodd 2 2 3 4 4 3 2 3 abeven 3 3 2 3 3 2 3 2 · Denote by D the set of partitions with parts colored with colors in {a, b, a2, b2, ab}, such that no part is equal to 1ab, 1a2 or 1b2, and consecutive differences at least equal to those in the table above.

Isaac KONAN Siladi` c’s partition theorem

7/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Partitions and Rogers-Ramanujan type Identities Siladi´ c’s partition theorem Dousse’s refinement

Minimal differences in terms of colored parts

λi\λi+1

a2

• dd

aodd aeven b2

• dd

bodd beven abodd abeven a2

• dd

4 4 3 4 4 3 4 3 aodd 2 2 3 2 2 3 2 1 aeven 3 3 2 3 3 2 3 2 b2

• dd

2 2 3 4 4 3 2 3 bodd 2 2 1 2 2 3 2 1 beven 1 1 2 3 3 2 1 2 abodd 2 2 3 4 4 3 2 3 abeven 3 3 2 3 3 2 3 2 · Denote by D the set of partitions with parts colored with colors in {a, b, a2, b2, ab}, such that no part is equal to 1ab, 1a2 or 1b2, and consecutive differences at least equal to those in the table above.

Isaac KONAN Siladi` c’s partition theorem

8/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Partitions and Rogers-Ramanujan type Identities Siladi´ c’s partition theorem Dousse’s refinement

Dousse’s refinement

Theorem (Dousse 2017)

Denote by D(u, v, n) the set of all the partitions of n in D, with u, v respectively equal to the number of occurrences of the color a, b. Denote by C(u, v, n) the set of all the partitions of n with respectively u, v distinct parts colored by a, b . We then have ♯D(u, v, n) = ♯C(u, v, n) · In terms of q-series, we get the following identity :

∞

• n=0

♯D(u, v, n)aubvqn =

∞

• n=1

(1 + aqn)(1 + bqn)

Isaac KONAN Siladi` c’s partition theorem

9/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Infinite set of primary colors Generalized theorem

Infinite ordered set of primary colors C

Set of colored parts with primary colors : P = N× × C.

Isaac KONAN Siladi` c’s partition theorem

9/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Infinite set of primary colors Generalized theorem

Infinite ordered set of primary colors C

colored part kc ≡ particle with energetic potential k and state c .

Isaac KONAN Siladi` c’s partition theorem

9/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Infinite set of primary colors Generalized theorem

Infinite ordered set of primary colors C

colored part kc ≡ particle with energetic potential k and state c .

Isaac KONAN Siladi` c’s partition theorem

9/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Infinite set of primary colors Generalized theorem

Infinite ordered set of primary colors C

colored part kc ≡ particle with energetic potential k and state c . Lexicographic strict order ≻ on P: kc ≻ k′

c′ ⇐

⇒ ≥

Isaac KONAN Siladi` c’s partition theorem

9/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Infinite set of primary colors Generalized theorem

Infinite ordered set of primary colors C

colored part kc ≡ particle with energetic potential k and state c . Lexicographic strict order ≻ on P: kc ≻ k′

c′ ⇐

⇒ k − k′ ≥ potential difference

Isaac KONAN Siladi` c’s partition theorem

9/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Infinite set of primary colors Generalized theorem

Infinite ordered set of primary colors C

colored part kc ≡ particle with energetic potential k and state c . Lexicographic strict order ≻ on P: kc ≻ k′

c′ ⇐

⇒ k − k′ ≥ χ(c ≤ c′) potential difference minimal energy where χ(A) = 1 if A is true and 0 if not.

Isaac KONAN Siladi` c’s partition theorem

9/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Infinite set of primary colors Generalized theorem

Infinite ordered set of primary colors C

colored part kc ≡ particle with energetic potential k and state c . Lexicographic strict order ≻ on P: kc ≻ k′

c′ ⇐

⇒ k − k′ ≥ χ(c ≤ c′) potential difference minimal energy where χ(A) = 1 if A is true and 0 if not. Example with • > • > •:

Isaac KONAN Siladi` c’s partition theorem

9/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Infinite set of primary colors Generalized theorem

Infinite ordered set of primary colors C

colored part kc ≡ particle with energetic potential k and state c . Lexicographic strict order ≻ on P: kc ≻ k′

c′ ⇐

⇒ k − k′ ≥ χ(c ≤ c′) potential difference minimal energy where χ(A) = 1 if A is true and 0 if not. Example with • > • > •: (3, •) = • ≻ • = (2, •)

Isaac KONAN Siladi` c’s partition theorem

9/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Infinite set of primary colors Generalized theorem

Infinite ordered set of primary colors C

colored part kc ≡ particle with energetic potential k and state c . Lexicographic strict order ≻ on P: kc ≻ k′

c′ ⇐

⇒ k − k′ ≥ χ(c ≤ c′) potential difference minimal energy where χ(A) = 1 if A is true and 0 if not. Example with • > • > •: (3, •) = • ≻ • = (2, •) (4, •) = • ≻ • = (4, •)

Isaac KONAN Siladi` c’s partition theorem

10/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Infinite set of primary colors Generalized theorem

Secondary colors

We define positive parts with secondary colors as elements in S = N× × C2 and sum of two consecutive parts with primary colors:

Isaac KONAN Siladi` c’s partition theorem

10/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Infinite set of primary colors Generalized theorem

Secondary colors

We define positive parts with secondary colors as elements in S = N× × C2 and sum of two consecutive parts with primary colors: (k, c, c′) = + =

Isaac KONAN Siladi` c’s partition theorem

10/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Infinite set of primary colors Generalized theorem

Secondary colors

We define positive parts with secondary colors as elements in S = N× × C2 and sum of two consecutive parts with primary colors: (k, c, c′) = (k + χ(c ≤ c′))c + (k)c′ = upper half lower half

Isaac KONAN Siladi` c’s partition theorem

10/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Infinite set of primary colors Generalized theorem

Secondary colors

We define positive parts with secondary colors as elements in S = N× × C2 and sum of two consecutive parts with primary colors: (k, c, c′) = (k + χ(c ≤ c′))c + (k)c′ = (2k + χ(c ≤ c′))cc′

Isaac KONAN Siladi` c’s partition theorem

10/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Infinite set of primary colors Generalized theorem

Secondary colors

We define positive parts with secondary colors as elements in S = N× × C2 and sum of two consecutive parts with primary colors: (k, c, c′) = (k + χ(c ≤ c′))c + (k)c′ = (2k + χ(c ≤ c′))cc′ potentials

Isaac KONAN Siladi` c’s partition theorem

10/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Infinite set of primary colors Generalized theorem

Secondary colors

We define positive parts with secondary colors as elements in S = N× × C2 and sum of two consecutive parts with primary colors: (k, c, c′) = (k + χ(c ≤ c′))c + (k)c′ = (2k + χ(c ≤ c′))cc′ states

Isaac KONAN Siladi` c’s partition theorem

10/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Infinite set of primary colors Generalized theorem

Secondary colors

We define positive parts with secondary colors as elements in S = N× × C2 and sum of two consecutive parts with primary colors: (k, c, c′) = (k + χ(c ≤ c′))c + (k)c′ = (2k + χ(c ≤ c′))cc′ Examples with • > • > •:

Isaac KONAN Siladi` c’s partition theorem

10/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Infinite set of primary colors Generalized theorem

Secondary colors

We define positive parts with secondary colors as elements in S = N× × C2 and sum of two consecutive parts with primary colors: (k, c, c′) = (k + χ(c ≤ c′))c + (k)c′ = (2k + χ(c ≤ c′))cc′ Examples with • > • > •: (2, •, •) = • + • = ••

Isaac KONAN Siladi` c’s partition theorem

10/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Infinite set of primary colors Generalized theorem

Secondary colors

We define positive parts with secondary colors as elements in S = N× × C2 and sum of two consecutive parts with primary colors: (k, c, c′) = (k + χ(c ≤ c′))c + (k)c′ = (2k + χ(c ≤ c′))cc′ Examples with • > • > •: (2, •, •) = • + • = •• (2, •, •) = • + • = ••

Isaac KONAN Siladi` c’s partition theorem

10/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Infinite set of primary colors Generalized theorem

Secondary colors

We define positive parts with secondary colors as elements in S = N× × C2 and sum of two consecutive parts with primary colors: (k, c, c′) = (k + χ(c ≤ c′))c + (k)c′ = (2k + χ(c ≤ c′))cc′ Examples with • > • > •: (2, •, •) = • + • = •• (2, •, •) = • + • = •• (2, •, •) = • + • = ••

Isaac KONAN Siladi` c’s partition theorem

11/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Infinite set of primary colors Generalized theorem

Formalizing Siladi´ c’s consecutive differences

We define relation ≫ on P ⊔ S as follows:

Isaac KONAN Siladi` c’s partition theorem

11/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Infinite set of primary colors Generalized theorem

Formalizing Siladi´ c’s consecutive differences

We define relation ≫ on P ⊔ S as follows:

potential difference, minimal energy,

• P × P: kc ≫ k′

c′ ⇐

⇒ k − k′ > χ(c ≤ c′) ,

Isaac KONAN Siladi` c’s partition theorem

11/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Infinite set of primary colors Generalized theorem

Formalizing Siladi´ c’s consecutive differences

We define relation ≫ on P ⊔ S as follows:

potential difference, minimal energy,

• P × P: kc ≫ k′

c′ ⇐

⇒ k − k′ > χ(c ≤ c′) ,

• P × S:

kc ≫ (k′, c′, c′′) ⇐ ⇒ k − (2k′ + χ(c′ ≤ c′′)) ≥ χ(c ≤ c′) + χ(c′ ≤ c′′) ,

• S × P:

(k, c, c′) ≫ k′

c′′ ⇐

⇒ (2k + χ(c ≤ c′)) − k′ > χ(c ≤ c′) + χ(c′ ≤ c′′) ,

Isaac KONAN Siladi` c’s partition theorem

11/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Infinite set of primary colors Generalized theorem

Formalizing Siladi´ c’s consecutive differences

We define relation ≫ on P ⊔ S as follows:

potential difference, minimal energy,

• P × P: kc ≫ k′

c′ ⇐

⇒ k − k′ > χ(c ≤ c′) ,

• P × S:

kc ≫ (k′, c′, c′′) ⇐ ⇒ k − (2k′ + χ(c′ ≤ c′′)) ≥ χ(c ≤ c′) + χ(c′ ≤ c′′) ,

• S × P:

(k, c, c′) ≫ k′

c′′ ⇐

⇒ (2k + χ(c ≤ c′)) − k′ > χ(c ≤ c′) + χ(c′ ≤ c′′) ,

• S × S:

(k, c, c′) ≫ (k′, c′′, c′′′) ⇐ ⇒ kc′ ≻ (k′ + χ(c′′ ≤ c′′′))c′′ ·

Isaac KONAN Siladi` c’s partition theorem

12/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Infinite set of primary colors Generalized theorem

Dousse’sdifferences with new color ba

Here we have a < b and the table

λi\λi+1

a2

• dd

aodd aeven b2

• dd

bodd beven abodd abeven a2

• dd

4 4 3 4 4 3 4 3 aodd 2 2 3 2 2 3 2 1 aeven 3 3 2 3 3 2 3 2 b2

• dd

2 2 3 4 4 3 2 3 bodd 2 2 1 2 2 3 2 1 beven 1 1 2 3 3 2 1 2 abodd 2 2 3 4 4 3 2 3 abeven 3 3 2 3 3 2 3 2 becomes

Isaac KONAN Siladi` c’s partition theorem

12/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Infinite set of primary colors Generalized theorem

Dousse’sdifferences with new color ba

Here we have a < b

λi\λi+1

a2 a b2 b ab ba a2 4 3 4 3 4 3 a 2 2 2 2 2 1 b2 2 2 4 3 2 3 b 1 1 2 2 1 1 ab 2 2 4 3 2 3 ba 3 2 3 2 3 2 and matches exactly with ≫.

Isaac KONAN Siladi` c’s partition theorem

13/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Infinite set of primary colors Generalized theorem

Generalized theorem

Theorem (K. 2018)

Let O = {partitions with ordered parts in (P, ≻)} and E = {partitions with ordered parts in (P ⊔ S, ≫)}. Then, for any fixed size n and color sequence C, there are as many partitions in O as partitions in E.

Isaac KONAN Siladi` c’s partition theorem

13/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Infinite set of primary colors Generalized theorem

Generalized theorem

Theorem (K. 2018)

Let O = {partitions with ordered parts in (P, ≻)} and E = {partitions with ordered parts in (P ⊔ S, ≫)}. Then, for any fixed size n and color sequence C, there are as many partitions in O as partitions in E.

Theorem (K. 2019’)

The previous theorem holds if we remplace the minimal energy χ(c ≤ c′) by any function ǫ : C2 → {0, 1} which satisfies the triangular inequality : ∀ c, c′, c′′ ∈ C , ǫ(c, c′′) ≤ ǫ(c, c′) + ǫ(c′, c′′) ·

Isaac KONAN Siladi` c’s partition theorem

14/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Infinite set of primary colors Generalized theorem

Direct corollaries

1 Analogous of Siladi´

c’s theorem : for {a, b} ⊂ C, a < b, ǫ(c, c′) = χ(c < c′) and (q, a, b) → (q4, q−3, q−1) , we get :

• dd parts ≡

λi − λi+1 λi + λi+1 mod 16 ±4 1 ±3 2 ±2, ±6 3 ±1, ±5, ±7 Isaac KONAN Siladi` c’s partition theorem

14/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Infinite set of primary colors Generalized theorem

Direct corollaries

1 Analogous of Siladi´

c’s theorem : for {a, b} ⊂ C, a < b, ǫ(c, c′) = χ(c < c′) and (q, a, b) → (q4, q−3, q−1) , we get :

• dd parts ≡

λi − λi+1 λi + λi+1 mod 16 ±4 1 ±3 2 ±2, ±6 3 ±1, ±5, ±7

2 Overpartitions with b > a > a > b and

ǫ(c, c′) = χ(c < c′) + χ(c = c′)χ(c is overlined) ·

Isaac KONAN Siladi` c’s partition theorem

14/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Infinite set of primary colors Generalized theorem

Direct corollaries

1 Analogous of Siladi´

c’s theorem : for {a, b} ⊂ C, a < b, ǫ(c, c′) = χ(c < c′) and (q, a, b) → (q4, q−3, q−1) , we get :

• dd parts ≡

λi − λi+1 λi + λi+1 mod 16 ±4 1 ±3 2 ±2, ±6 3 ±1, ±5, ±7

2 Overpartitions with b > a > a > b and

ǫ(c, c′) = χ(c < c′) + χ(c = c′)χ(c is overlined) ·

3 Euler distinct-odd : Color a, kth iteration of theorem with

primary color a2k−1 and ǫ(a2k−1, a2k−1) = 0, and dilation (q, a) → (q2, q−1).

Isaac KONAN Siladi` c’s partition theorem

15/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

Transfer function

Inspired by BRESSOUD’s bijective proof of Schur’s 1926 partition theorem (1980)

Transfer function Γ on P × S ⊔ S × P:

Isaac KONAN Siladi` c’s partition theorem

15/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

Transfer function

Inspired by BRESSOUD’s bijective proof of Schur’s 1926 partition theorem (1980)

Transfer function Γ on P × S ⊔ S × P: (k, c), (k′, c′, c′′) → (k′ , c, c′), (k , c′′)

Isaac KONAN Siladi` c’s partition theorem

15/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

Transfer function

Inspired by BRESSOUD’s bijective proof of Schur’s 1926 partition theorem (1980)

Transfer function Γ on P × S ⊔ S × P: (k, c), (k′, c′, c′′) → (k′+ǫ(c′, c′′), c, c′), (k−ǫ(c, c′)−ǫ(c′, c′′), c′′)

Isaac KONAN Siladi` c’s partition theorem

15/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

Transfer function

Inspired by BRESSOUD’s bijective proof of Schur’s 1926 partition theorem (1980)

Transfer function Γ on P × S ⊔ S × P: (k, c), (k′, c′, c′′) → (k′+ǫ(c′, c′′), c, c′), (k−ǫ(c, c′)−ǫ(c′, c′′), c′′) (k, c, c′), (k′, c′′) → (k′+ǫ(c, c′)+ǫ(c′, c′′), c), (k−ǫ(c′, c′′), c′, c′′)

Isaac KONAN Siladi` c’s partition theorem

15/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

Transfer function

Inspired by BRESSOUD’s bijective proof of Schur’s 1926 partition theorem (1980)

Transfer function Γ on P × S ⊔ S × P: (k, c), (k′, c′, c′′) → (k′+ǫ(c′, c′′), c, c′), (k−ǫ(c, c′)−ǫ(c′, c′′), c′′) (k, c, c′), (k′, c′′) → (k′+ǫ(c, c′)+ǫ(c′, c′′), c), (k−ǫ(c′, c′′), c′, c′′) Example with • > • > • and ǫ(c, c′) = χ(c ≤ c′):

Isaac KONAN Siladi` c’s partition theorem

15/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

Transfer function

Inspired by BRESSOUD’s bijective proof of Schur’s 1926 partition theorem (1980)

Transfer function Γ on P × S ⊔ S × P: (k, c), (k′, c′, c′′) → (k′+ǫ(c′, c′′), c, c′), (k−ǫ(c, c′)−ǫ(c′, c′′), c′′) (k, c, c′), (k′, c′′) → (k′+ǫ(c, c′)+ǫ(c′, c′′), c), (k−ǫ(c′, c′′), c′, c′′) Example with • > • > • and ǫ(c, c′) = χ(c ≤ c′): (8, •), (2, •, •) ↔ (3, •, •), (7, •) , ↔ ,

Isaac KONAN Siladi` c’s partition theorem

15/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

Transfer function

Inspired by BRESSOUD’s bijective proof of Schur’s 1926 partition theorem (1980)

Transfer function Γ on P × S ⊔ S × P: (k, c), (k′, c′, c′′) → (k′+ǫ(c′, c′′), c, c′), (k−ǫ(c, c′)−ǫ(c′, c′′), c′′) (k, c, c′), (k′, c′′) → (k′+ǫ(c, c′)+ǫ(c′, c′′), c), (k−ǫ(c′, c′′), c′, c′′) Example with • > • > • and ǫ(c, c′) = χ(c ≤ c′): (8, •), (2, •, •) ↔ (3, •, •), (7, •) , ↔ , (2, •), (3, •, •) ↔ (3, •, •), (1, •) , ↔ ,

Isaac KONAN Siladi` c’s partition theorem

16/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

From O to E

Example with (14, 9, 7, 6, 5, 4, 3, 3, 3)

• Isaac KONAN

Siladi` c’s partition theorem

16/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

From O to E

Example with (14, 9, 7, 6, 5, 4, 3, 3, 3)

Eros

• Isaac KONAN

Siladi` c’s partition theorem

16/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

From O to E

Example with (14, 9, 7, 6, 5, 4, 3, 3, 3)

Eros

• ∗
• ∗
• 1 By beginning with the greatest potentials, sum up consecutive

troublesome pair (λi, λi+1), (which are too close by ≻, so that we don’t have ≫).

Isaac KONAN Siladi` c’s partition theorem

16/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

From O to E

Example with (14, 9, 7, 6, 5, 4, 3, 3, 3)

Eros

• ∗
• ∗
• 1 By beginning with the greatest potentials, sum up consecutive

troublesome pair (λi, λi+1), (which are too close by ≻, so that we don’t have ≫).

Isaac KONAN Siladi` c’s partition theorem

16/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

From O to E

Example with (14, 9, 7, 6, 5, 4, 3, 3, 3)

Eros

• ∗
• ∗

1 By beginning with the greatest potentials, sum up consecutive

troublesome pair (λi, λi+1), (which are too close by ≻, so that we don’t have ≫).

Isaac KONAN Siladi` c’s partition theorem

16/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

From O to E

Example with (14, 9, 7, 6, 5, 4, 3, 3, 3)

Eros

• 1 By beginning with the greatest potentials, sum up consecutive

troublesome pair (λi, λi+1), (which are too close by ≻, so that we don’t have ≫).

Isaac KONAN Siladi` c’s partition theorem

16/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

From O to E

Example with (14, 9, 7, 6, 5, 4, 3, 3, 3)

Eros

• ∗
• •∗
• 1 By beginning with the greatest potentials, sum up consecutive

troublesome pair (λi, λi+1), (which are too close by ≻, so that we don’t have ≫).

2 As long as there is a pair (λi, λi+1) ∈ P × S such that

λi ≫ λi+1, replace it by Γ(λi, λi+1).

Isaac KONAN Siladi` c’s partition theorem

16/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

From O to E

Example with (14, 9, 7, 6, 5, 4, 3, 3, 3)

Eros

• ∗
• •∗
• 1 By beginning with the greatest potentials, sum up consecutive

troublesome pair (λi, λi+1), (which are too close by ≻, so that we don’t have ≫).

2 As long as there is a pair (λi, λi+1) ∈ P × S such that

λi ≫ λi+1, replace it by Γ(λi, λi+1).

Isaac KONAN Siladi` c’s partition theorem

16/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

From O to E

Example with (14, 9, 7, 6, 5, 4, 3, 3, 3)

Eros

• ∗
• •∗
• 1 By beginning with the greatest potentials, sum up consecutive

troublesome pair (λi, λi+1), (which are too close by ≻, so that we don’t have ≫).

2 As long as there is a pair (λi, λi+1) ∈ P × S such that

λi ≫ λi+1, replace it by Γ(λi, λi+1).

Isaac KONAN Siladi` c’s partition theorem

16/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

From O to E

Example with (14, 9, 7, 6, 5, 4, 3, 3, 3)

Eros

• ∗
• •∗

1 By beginning with the greatest potentials, sum up consecutive

troublesome pair (λi, λi+1), (which are too close by ≻, so that we don’t have ≫).

2 As long as there is a pair (λi, λi+1) ∈ P × S such that

λi ≫ λi+1, replace it by Γ(λi, λi+1).

Isaac KONAN Siladi` c’s partition theorem

16/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

From O to E

Example with (14, 9, 7, 6, 5, 4, 3, 3, 3)

Eros

• 1 By beginning with the greatest potentials, sum up consecutive

troublesome pair (λi, λi+1), (which are too close by ≻, so that we don’t have ≫).

2 As long as there is a pair (λi, λi+1) ∈ P × S such that

λi ≫ λi+1, replace it by Γ(λi, λi+1).

Isaac KONAN Siladi` c’s partition theorem

16/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

From O to E

Example with (14, 9, 7, 6, 5, 4, 3, 3, 3)

Eros

• 1 By beginning with the greatest potentials, sum up consecutive

troublesome pair (λi, λi+1), (which are too close by ≻, so that we don’t have ≫).

2 As long as there is a pair (λi, λi+1) ∈ P × S such that

λi ≫ λi+1, replace it by Γ(λi, λi+1).

3 The final result is our image in E :

(14, 7 + 7, 5 + 4, 7, 3 + 3, 4) ·

Isaac KONAN Siladi` c’s partition theorem

17/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

From E to O

Example with 8 + 7, 12, 10, 5 + 5, 4 + 3, 5, 4

• Isaac KONAN

Siladi` c’s partition theorem

17/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

From E to O

Example with 8 + 7, 12, 10, 5 + 5, 4 + 3, 5, 4

Eris

• Isaac KONAN

Siladi` c’s partition theorem

17/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

From E to O

Example with 8 + 7, 12, 10, 5 + 5, 4 + 3, 5, 4

Eris

• •∗
• ∗
• 1 As long as there is a pair (λi, λi+1) ∈ S × P such that

lowerhalf (λi) ≻ λi+1, replace it by Γ(λi, λi+1).

Isaac KONAN Siladi` c’s partition theorem

17/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

From E to O

Example with 8 + 7, 12, 10, 5 + 5, 4 + 3, 5, 4

Eris

• •∗
• ∗

1 As long as there is a pair (λi, λi+1) ∈ S × P such that

lowerhalf (λi) ≻ λi+1, replace it by Γ(λi, λi+1).

Isaac KONAN Siladi` c’s partition theorem

17/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

From E to O

Example with 8 + 7, 12, 10, 5 + 5, 4 + 3, 5, 4

Eris

• •∗
• ∗
• 1 As long as there is a pair (λi, λi+1) ∈ S × P such that

lowerhalf (λi) ≻ λi+1, replace it by Γ(λi, λi+1).

Isaac KONAN Siladi` c’s partition theorem

17/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

From E to O

Example with 8 + 7, 12, 10, 5 + 5, 4 + 3, 5, 4

Eris

• •∗
• ∗
• 1 As long as there is a pair (λi, λi+1) ∈ S × P such that

lowerhalf (λi) ≻ λi+1, replace it by Γ(λi, λi+1).

Isaac KONAN Siladi` c’s partition theorem

17/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

From E to O

Example with 8 + 7, 12, 10, 5 + 5, 4 + 3, 5, 4

Eris

• •∗
• ∗
• 1 As long as there is a pair (λi, λi+1) ∈ S × P such that

lowerhalf (λi) ≻ λi+1, replace it by Γ(λi, λi+1).

Isaac KONAN Siladi` c’s partition theorem

17/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

From E to O

Example with 8 + 7, 12, 10, 5 + 5, 4 + 3, 5, 4

Eris

• •∗
• ∗
• 1 As long as there is a pair (λi, λi+1) ∈ S × P such that

lowerhalf (λi) ≻ λi+1, replace it by Γ(λi, λi+1).

Isaac KONAN Siladi` c’s partition theorem

17/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

From E to O

Example with 8 + 7, 12, 10, 5 + 5, 4 + 3, 5, 4

Eris

• •∗
• ∗
• 1 As long as there is a pair (λi, λi+1) ∈ S × P such that

lowerhalf (λi) ≻ λi+1, replace it by Γ(λi, λi+1).

Isaac KONAN Siladi` c’s partition theorem

17/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

From E to O

Example with 8 + 7, 12, 10, 5 + 5, 4 + 3, 5, 4

Eris

• •∗
• ∗
• 1 As long as there is a pair (λi, λi+1) ∈ S × P such that

lowerhalf (λi) ≻ λi+1, replace it by Γ(λi, λi+1).

Isaac KONAN Siladi` c’s partition theorem

17/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

From E to O

Example with 8 + 7, 12, 10, 5 + 5, 4 + 3, 5, 4

Eris

• 1 As long as there is a pair (λi, λi+1) ∈ S × P such that

lowerhalf (λi) ≻ λi+1, replace it by Γ(λi, λi+1).

Isaac KONAN Siladi` c’s partition theorem

17/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

From E to O

Example with 8 + 7, 12, 10, 5 + 5, 4 + 3, 5, 4

Eris

• •∗
• 1 As long as there is a pair (λi, λi+1) ∈ S × P such that

lowerhalf (λi) ≻ λi+1, replace it by Γ(λi, λi+1).

2 In the final result, split parts in S into their upper and lower

halves.

Isaac KONAN Siladi` c’s partition theorem

17/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

From E to O

Example with 8 + 7, 12, 10, 5 + 5, 4 + 3, 5, 4

Eris

• •∗
• 1 As long as there is a pair (λi, λi+1) ∈ S × P such that

lowerhalf (λi) ≻ λi+1, replace it by Γ(λi, λi+1).

2 In the final result, split parts in S into their upper and lower

halves.

Isaac KONAN Siladi` c’s partition theorem

17/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

From E to O

Example with 8 + 7, 12, 10, 5 + 5, 4 + 3, 5, 4

Eris

• •∗

1 As long as there is a pair (λi, λi+1) ∈ S × P such that

lowerhalf (λi) ≻ λi+1, replace it by Γ(λi, λi+1).

2 In the final result, split parts in S into their upper and lower

halves.

Isaac KONAN Siladi` c’s partition theorem

17/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

From E to O

Example with 8 + 7, 12, 10, 5 + 5, 4 + 3, 5, 4

Eris

• 1 As long as there is a pair (λi, λi+1) ∈ S × P such that

lowerhalf (λi) ≻ λi+1, replace it by Γ(λi, λi+1).

2 In the final result, split parts in S into their upper and lower

halves.

Isaac KONAN Siladi` c’s partition theorem

17/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

From E to O

Example with 8 + 7, 12, 10, 5 + 5, 4 + 3, 5, 4

Eris

• 1 As long as there is a pair (λi, λi+1) ∈ S × P such that

lowerhalf (λi) ≻ λi+1, replace it by Γ(λi, λi+1).

2 In the final result, split parts in S into their upper and lower

halves.

3 The obtained result is our image in O:

(14, 12, 8, 6, 5, 5, 4, 3, 3, 3) ·

Isaac KONAN Siladi` c’s partition theorem

18/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

Why does it work?

Isaac KONAN Siladi` c’s partition theorem

18/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

Why does it work?

Because of these two main properties of Γ : For any (p, s) ∈ P × S, and (s′, p′) = Γ(p, s),

1 p ≫ s ⇐

⇒ s′ ≫ p′.

2 p ≫ upperhalf (s) ⇐

⇒ lowerhalf (s′) ≻ p′.

Isaac KONAN Siladi` c’s partition theorem

18/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

Why does it work?

Because of these two main properties of Γ : For any (p, s) ∈ P × S, and (s′, p′) = Γ(p, s),

1 p ≫ s ⇐

⇒ s′ ≫ p′.

2 p ≫ upperhalf (s) ⇐

⇒ lowerhalf (s′) ≻ p′.

• Uniqueness of the definition of troublesome pairs of parts

Isaac KONAN Siladi` c’s partition theorem

18/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

Why does it work?

Because of these two main properties of Γ : For any (p, s) ∈ P × S, and (s′, p′) = Γ(p, s),

1 p ≫ s ⇐

⇒ s′ ≫ p′.

2 p ≫ upperhalf (s) ⇐

⇒ lowerhalf (s′) ≻ p′.

• Uniqueness of the definition of troublesome pairs of parts
• Transfer function moves parts with minimal energy, so that

they always stay positive.

Isaac KONAN Siladi` c’s partition theorem

18/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

Why does it work?

Because of these two main properties of Γ : For any (p, s) ∈ P × S, and (s′, p′) = Γ(p, s),

1 p ≫ s ⇐

⇒ s′ ≫ p′.

2 p ≫ upperhalf (s) ⇐

⇒ lowerhalf (s′) ≻ p′.

• Uniqueness of the definition of troublesome pairs of parts
• Transfer function moves parts with minimal energy, so that

they always stay positive.

• There exist unique functions on P × S, related to potential

differences p − s and lowerhalf (s) − p, that do not depend on the steps of the applications of Γ.

Isaac KONAN Siladi` c’s partition theorem

19/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

Parts of degree more than 2 and difference relations

Isaac KONAN Siladi` c’s partition theorem

19/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

Parts of degree more than 2 and difference relations

• Parts of c1 · · · ck as sum of k consecutive parts with primary

colors...

Isaac KONAN Siladi` c’s partition theorem

19/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

Parts of degree more than 2 and difference relations

• Parts of c1 · · · ck as sum of k consecutive parts with primary

colors...

• Difference relation ≫ according to the sum of minimal

energies to switch parts of degrees k, k′.

Isaac KONAN Siladi` c’s partition theorem

19/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

Parts of degree more than 2 and difference relations

• Parts of c1 · · · ck as sum of k consecutive parts with primary

colors...

• Difference relation ≫ according to the sum of minimal

energies to switch parts of degrees k, k′.

• Same transfer function with the same properties according to

≫ and ≻.

Isaac KONAN Siladi` c’s partition theorem

19/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees

Parts of degree more than 2 and difference relations

• Parts of c1 · · · ck as sum of k consecutive parts with primary

colors...

• Difference relation ≫ according to the sum of minimal

energies to switch parts of degrees k, k′.

• Same transfer function with the same properties according to

≫ and ≻.

• Most likely conjecture : less partitions in O than partitions in

E+.

Isaac KONAN Siladi` c’s partition theorem

20/20 A Rogers-Ramanujan type identy : Siladi´ c’s partition theorem Generalization of Siladi´ c’s theorem Bijective map Further analysis on higher degrees Isaac KONAN Siladi` c’s partition theorem